Constructions of -regular maps using finite local schemes

  • Jarosław Buczyński

    University of Warsaw and Polish Academy of Sciences, Warsaw, Poland
  • Tadeusz Januszkiewicz

    Polish Academy of Sciences, Warsaw, Poland
  • Joachim Jelisiejew

    University of Warsaw, Poland
  • Mateusz Michałek

    Freie Universität Berlin, Germany, and Polish Academy of Sciences, Warsaw, Poland
Constructions of $k$-regular maps using finite local schemes cover
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Abstract

A continuous map or is called -regular if the images of any points are linearly independent. Given integers and a problem going back to Chebyshev and Borsuk is to determine the minimal value of for which such maps exist. The methods of algebraic topology provide lower bounds for , but there are very few results on the existence of such maps for particular values and . Using methods of algebraic geometry we construct -regular maps. We relate the upper bounds on with the dimension of the locus of certain Gorenstein schemes in the punctual Hilbert scheme. The computations of the dimension of this family is explicit for , and we provide explicit examples for . We also provide upper bounds for arbitrary and .

Cite this article

Jarosław Buczyński, Tadeusz Januszkiewicz, Joachim Jelisiejew, Mateusz Michałek, Constructions of -regular maps using finite local schemes. J. Eur. Math. Soc. 21 (2019), no. 6, pp. 1775–1808

DOI 10.4171/JEMS/873